Notes on Chapter 2 of the Logic Manual by Volker Halbach The chapter covers the syntax and semantics of propositional logic Written by a student at Oxford University achieving first class grades in logic This contains all definitions and relevant material from the original text.
2. Syntax and Semantics of Propositional
Logic
2.1: Quotation
Quotation is used to refer to specific expressions
This can be to specific language or to metavariables (metalinguistic variables)
E.g. If 𝜑 and 𝜓 are English sentences then '𝜑 and 𝜓' is an English sentence
Since expressions of the formal languages logicians are concerned with often differ from
English, they usually drop the quotation marks
E.g. (P→(𝑄 ∧ 𝑅)) is a sentence of the language of propositional logic
2.2: The Syntax of the Language of Propositional Logic
Definition 2.1
P, Q, R, P1, Q1, R1, P2.... Are sentence letters
Definition 2.2
i. All sentence letters are sentences of L1
ii. If 𝝋 and 𝝍 are sentences of L1 then ¬𝝋, (𝝋 ∧ 𝝍), (𝝋 ∨ 𝝍), (𝝋 → 𝝍) and
(𝝋 ↔ 𝝍) are sentences of L1
iii. Nothing else is a sentence of L1
2.3: Rules for Dropping Brackets
Bracketing Convention 1
, The outer brackets may be omitted from a sentence that is not part of another sentence
E.g. P→(𝑄 ∨ 𝑅) as opposed to (P→(𝑄 ∨ 𝑅))
N.B. This doesn't apply to sentences that are negated e.g. ¬(P→(𝑄 ∨ 𝑅))
Because (P→(𝑄 ∨ 𝑅)) is part of the sentence ¬(P→(𝑄 ∨ 𝑅))
Bracketing convention 2
The inner set of brackets may be omitted from a sentence of the form ((𝝋 ∧ 𝝍) ∧ 𝝌). An
analogous convention applies to ∨
E.g. ((𝑃 ∧ 𝑄) ∧ 𝑅) can be abbreviated as (P ∧ 𝑄 ∧ 𝑅) which by convention 1 can also be
abbreviated as P ∧ 𝑄 ∧ 𝑅
Just as x binds more strongly than + in arithmetic:
∧ and ∨ bind more strongly than → or ↔
Hence:
Bracketing convention 3
Assume 𝝋, 𝝍, and 𝝌 are sentences in L1, ∗ is either ∧ or ∨, and ∘ is either → or ↔. Then,
if (𝝋 ∘ (𝝍 ∗ 𝝌)) or ((𝝋 ∗ 𝝍) ∘ 𝝌) occurs as part of the sentence that is to be abbreviated,
the inner set of brackets may be omitted.
2.4: The Semantics of Propositional Logic
The only parts of language L1 that can be interpreted in different ways are the non-logical
symbols of L1: the sentence letters
Interpretations are provided by L1-structures which only need provide enough information to
determine whether a sentence is true or false
E.g. P ∧ Q is only true if both P and Q are true
If a sentence is true it has the truth-value True (T), and if it is false it has the truth-value False
(F)
Definition 2.5
An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence
letter of L1
E.g.
Starting from the truth-values assigned to the sentence letters by an L1-structure, one can work
out the truth-values for sentences containing connectives.
E.g. If P has truth value T and so does Q1 then P ∧ Q1 would receive the truth value T, too
Definition 2.6
Let A be some L1-structure, then |… |A assigns either T or F to every sentence of L1 in the
following way.
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